In the last fifteen years, there have been many beautiful applications given by interpreting orbit spaces of representations as moduli spaces of arithmetic or algebraic objects, such as ideal classes of low-rank rings or Selmer elements of elliptic curves. Many of the representations that arise seem to be closely related to one another, and in some cases, they can be formally related to one another by a process called Hermitianization; see [1, 2, 11]. In this paper, we construct such representations via a uniform approach. Our method relies on a seemingly unrelated problem: defining Galois closures of possibly noncommutative rings.
2 W. Ho and M. Satriano Galois closures of commutative rank n ring extensions were studied in [5], building on previous work of Grothendieck [6, Exposé 4], Katz-Mazur [9, Section 1.8.2],
and Gabber [7, Section 5.2]. Given a morphism R → A of commutative rings realizing A as a free R-module of rank n, the Galois closure G(A/R) is defined as the quotient A ⊗n /I A/R , where I A/R is an ideal generated by relations coming from characteristic polynomials. More precisely, given a ∈ A, consider the R-linear endomorphism of A given by multiplication by a and let T n + n j=1 (-1) j s A,j (x)T n-j be its characteristic polynomial. Let a (i)
where a is in the i-th tensor factor, and let e j denote the j-th elementary symmetric function. Then the ideal I A/R is generated by the relations e j a (1) , a (2) , . . . , a (n)s A,j (a), as a runs through all elements of A. Then G(A/R) is an R-algebra equipped with a natural S n -action, and the elements a (1) , a (2) , . . . , a (n) behave as if they are "Galois conjugates." One key property is that G(A/R) commutes with base change on R. This construction has since been generalized by Gioia to so-called intermediate Galois closure [8] as well as by Biesel to Galois closures associated to subgroups of S n [3].
We now describe the connection between Galois closures of noncommutative algebras and problems in arithmetic invariant theory. In this paper, we obtain many of the representations with arithmetic applications by the following uniform construction: let A be a possibly noncommutative degree n R-algebra and let G(A/R) be its Galois closure, as we define in Section 2, which comes with a natural S n -action. For an ndimensional m × m × • • • × m array with entries in G(A/R), there are two natural S nactions: one on G(A/R) and the other permuting the coordinates of the n-dimensional array. The subspace where these two actions coincide has a natural action of the matrix ring Mat m (A) ⊗ G(A/R); we refer to this as the associated Hermitian representation H A,m . See Section 4 for lists of representations obtained in this manner that have arisen in arithmetic invariant theory. We hope that our uniform construction of these representations H A,m , with just the input of a degree n R-algebra A and a positive integer m, will also give a systematic approach to studying the moduli problems related to the orbit spaces of the Hermitian representations.
For n and m sufficiently small, these Hermitian representations were studied explicitly in previous work [1, 2, 11]. Galois closures were not needed in these previous papers for two main reasons. First, when m is small, the entries of the elements in the Hermitian representation may be defined over A itself. And second, when n is small, the Galois closure G(A/R) is quite simple; for example, when Downloaded from https://academic.oup.com/imrn/article-abstract/doi/10.1093/imrn/rny231/5114641 by guest on 07 July 2020
A is a quadratic algebra (n = 2), the Galois closure G(A/R) is isomorphic to A (see Proposition 3.2). Similarly, when A is a decomposable cubic algebra, for example, The aforementioned Proposition 3.6 and Example 4.7 are both specific cases of more general results as we now discuss. In Section 2.3, we prove the following product formula, which allows one to calculate the Galois closure of decomposable algebras in terms of the Galois closures of its components.
where N is the multinomial coefficient
As a consequence, in Theorem 4.5, we may write the Hermitian representation of a decomposable algebra in terms of the Hermitian representations of its components.
Theorem (Product formula for Hermitianizations). For 1 ≤ i ≤ k, let A i be a degree n i R-algebra, and let A = k i=1 A i . Then for any positive integer m, we have
We also prove that taking Galois closures commutes with base change.
Let A be a degree n R-algebra and let S be a commutative
In this paper, all the algebras we consider are associative, but we believe it would be useful to generalize these ideas to non-associative algebras such as cubic Jordan algebras as well, especially as Jordan algebras have already been crucially used in basic examples of Hermitianization.
In this section, we define the Galois closure for certain classes of (possibly noncommutative) rings and discuss several properties. When A is commutative, it recovers the construction in [5].
We first define a degree n algebra over a commutative ring R. We use the following
Let A be a central R-algebra that is free of finite rank as an R-module.
Let R be a finitely generated commutative R-algebra such that R → R is universally injective, with a universally injective R-algebra homomorphism ι : A → Mat n (R ). We say that the triple (A, R , ι) is a degree n R-algebra if for all a ∈ A, the characteristic polynomial
We frequently suppress R and ι from the notation if they are unambiguous and refer to A itself as a degree n R-algebra.
For a degree n R-algebra (A, R , ι), we refer to Tr(a) := s A,1 (a) and N(a) := s A,n (a)
as the trace and norm of a ∈ A, respectively. It is immediate from the definition that s A,j (ra) = r j s A,j (a) for all r ∈ R and a ∈ A; in particular, the trace is additive and the norm is multiplicative.
conjugation by an element of GL n (R ). Then (A, R , ψι) is a degree n R-algebra and all the characteristic polynomials P A,a (T) for the two algebras coincide.
Let A be a central R-algebra that is free of finite rank n as an R-module.
If A = Mat n (R), then taking ι to be the identity map gives A the structure of a degree n algebra.
If A is a central simple algebra over a field F, then there exists a splitting field K over F such that A ⊗ F K Mat n (K), where n is the square root of the rank of A as a F-vector space. We thus have an injection ι : A → Mat n (K), and it is universally injective because ι is split. The polynomials P A,a agree with the reduced characteristic polynomial of a central simple algebra A, and it is well known that the coefficients lie in F (see, e.g., [4, Section IV.2]). In this way, we may view A as an algebra of degree equal to the square root of the rank of A (which is the typical definition of the degree of a central simple algebra).
Remark 2.10. One may generalize the definition of degree n algebra to locally free R-modules of finite rank by requiring a universally injective homomorphism to an endomorphism algebra of a rank n vector bundle over R instead. The definitions of and theorems for Galois closures will also generalize in a similar way, but we focus on the case of free R-modules in the rest of the paper for simplicity.
We next introduce products of degree n R-algebras.
Definition 2.11. Let (A 1 , R 1 , ι 1 ) and (A 2 , R 2 , ι 2 ) be R-algebras of degrees n 1 and n 2 , respectively. We define the product
where the last injection is given by block diagonals. If a i ∈ A i has characteristic
The following gives some further properties of product R-algebras.
Lemma 2.12. Let A i be a degree n i R-algebra for 1 ≤ i ≤ k and endow A = k i=1 A i with its associated structure as a degree n := k i=1 n i R-algebra.
where we set s A i ,0 (a i ) = 1.
2. If a = (0, . . . , 0, a j , 0, . . . 0), then
Proof. We first show (1). We use the notation [T j ]Q to denote the T j -coefficient of a polynomial Q(T). As P A,a (T) = j P A j ,a j (T), we have
Since the indices of the summation satisfy 0 ≤ m i ≤ n i , replacing m i by n im i changes the above sum to
and multiplying by (-1) m = i (-1) m i gives the result. For (2), we know that for all i = j, we have
Thus, by (1) we see s A,m (a) = s A j ,m (a j ).
We now define the Galois closure for a degree n R-algebra A. For an element a ∈ A, let
, where a is in the i-th tensor factor. Then consider the left ideal I A/R in A ⊗n generated by the elements e j (a (1) , a (2) , . . . ,
for every a ∈ A and 1 ≤ j ≤ n, where e j denotes the j-th elementary symmetric function and
Definition 2.14. The Galois closure of a degree n R-algebra A is defined to be the left Remark 2.17. Unlike the case of commutative rings considered in [5], here G(A/R) does not necessarily have a natural ring structure since I A/R is not necessarily a two-sided ideal. In fact, in many cases of interest (e.g., if A is the ring of n × n matrices Mat n (R)
for n ≥ 3), if we were to replace I A/R by the two-sided ideal generated by the elements (2.13), the expression (2.15) would become 0.
Remark 2.18. If (A, R , ι) is a degree n algebra and B is an R-algebra with a universally injective homomorphism B → A, then (B, R , ι) inherits the structure of a degree n algebra. Then since a degree n R-algebra. In this case, the characteristic polynomials P A,a for (A, R , ι) and (A, R , ψι) are the same, so the associated Galois closures agree.
By definition, the module G(A/R) has n distinct A-actions (one on each tensor factor). We denote the i-th action of a ∈ A on an element b ∈ G(A/R) as a• i b. Furthermore, the natural action of S n on A ⊗n induces an S n -action on G(A/R) with the following property: for all σ ∈ S n , a ∈ A, and b ∈ G(A/B), we have σ (a
gives G(A/R) the structure of a left module over the twisted group ring A ⊗n * S n (or equivalently, an S n -equivariant left A ⊗n -module).
The focus of this subsection is to prove two main properties of Galois closures: they commute with base change and they satisfy a product formula. Our 1st step is to show that the left ideal I A/R is generated by the expressions (2.13) for basis elements.
Proposition 2.20. Let A be a degree n R-algebra. If u 1 , . . . , u m is a basis for A over R, then I A/R is the left A ⊗n -ideal generated by the expressions e j a (1) , a (2) , . . . , a (n)s j (a)
for a ∈ {u 1 , . . . , u m }.
Proof. By definition, there is a finitely generated commutative R-algebra R and a universally injective R-algebra morphism ι :
As shown in [5, Lemma 11], for a noncommutative polynomial ring Z X, Y over Z generated by X and Y, there is a unique sequence
In particular, for any a, b ∈ A, letting x = ι(a) and y = ι(b), we have for i < k, where the q j are noncommutative polynomials. In the case where A is the split degree n algebra R n as in Example 2.6, the s A,k are the elementary symmetric functions e k .
Combining the above with the observation that s A,i (ra) = r i s A,i (a) for all r ∈ R, we see by induction on k that s A,k (a) is expressible in terms of the s A,j (u ) for j ≤ k.
Since the elementary symmetric functions e k satisfy these same relations (since they correspond to the special case where A = R n ), we conclude that I A/R is generated by the expressions e j (a (1) , a (2) , . . . , a (n) )s j (a) for a ∈ {u 1 , . . . , u m }.
Next, we show that if A is a degree n R-algebra, and R → S is a map of commutative rings, then A ⊗ R S carries a natural degree n S-algebra structure.
Lemma 2.23. Let R be a commutative ring and S a commutative R-algebra.
Let A be a degree n R-algebra and let S be a commutative R-algebra. The base change map yields an isomorphism Proof. Let j (a) := e j (a (1) , a (2) , . . . , a (n) )-s j (a). If u 1 , . . . , u m is a basis for A over R, then the u ⊗1 gives a basis for A⊗ R S over S. Proposition 2.20 implies that I A/R is generated by the expressions j (u ) and I (A⊗ R S)/S is generated by the expressions j (u ⊗ 1) = j (u )⊗ 1.
Hence, I (A⊗ R S)/S is the extension of the ideal I A/R . Consequently, the base change map is
We next compute the Galois closure of a product in terms of the Galois closures of the factors.
Theorem 2.26 (Product formula). For 1 ≤ j ≤ k, let A j be a degree n j R-algebra and
where N is the multinomial coefficient
), and the A ⊗n -action on the right-hand side of (2.27) is given by n i actions of A on each G(A i /R).
Proof. Since A = j A j , there exist idempotents ε j in the center Z(A) of A for 1 ≤ j ≤ k, such that A j = ε j A = Aε j and ε i ε j = δ i,j ε j , where δ is the Kronecker delta function. Let
[k] = {1, 2, • • • , k} and for every n-tuple i
Then
This product decomposition corresponds to the idempotents ε i ∈ Z(A ⊗n ) defined by
Notice that if J ⊆ A ⊗n is a left ideal, then ε i J = Jε i is a left ideal, which can be identified with a left ideal J i of A i . Moreover, J = i J i . In particular,
. Let i be an n-tuple for which this does not hold; then there is some j with #{ | i = j} < n j . By Lemma 2.12 (2), we know that s A,n j (ε j ) = s A j ,n j (1) = 1. So, the n j -th elementary symmetric function in the ε ( )
Since j occurs fewer than n j times in the n-tuple i, we see
= 0 for every summand above, and so
. We show in this case that
To do so, it is enough to consider the specific case where i
), since this is the case up to permutation. First note that A is generated by elements of the form a j ε j where a j ∈ A j and 1 ≤ j ≤ n. So, by Theorem 2.25, we know that I A ⊗n /R is generated as a left ideal by elements of the form
and so I i is generated by the above elements after left multiplying by ε i . First notice that by Lemma 2.12 (2) we know s A,m (a j ε j ) = s A j ,m (a j ), which is 0 if m > n j . Next note that i = j if and only if
So, multiplying the above expression by ε i is 0 if m > n j , and otherwise we obtain
that is nothing more than
This shows
As in Example 2.7, for any n ≥ 1, we may view the R-algebra A = R as a degree n algebra with characteristic polynomial P R,r (T) = (Tr) n . It is easy to check that I A/R = 0, so the Galois closure G(A/R) is isomorphic to R itself. The S n -action is trivial, and the n A-actions are all the same, namely the usual multiplication by elements of A = R. This seemingly trivial example plays a role in many of the examples in Section 4.4.
Suppose A is a degree 2 R-algebra. Then every element a ∈ A satisfies an equation of the for all a ∈ A. We will show that G(A/R) is isomorphic to A in this case. We first give
A the structure of a left (A ⊗2 * S 2 )-module as follows: given b ∈ A and a pure tensor Proof. One easily checks that ϕ is well defined and a morphism of left (A ⊗2 * S 2 )modules. It is clear that ϕ(I A/R ) = 0, so we obtain an induced map ϕ : G(A/R) → A. Note that ϕ(a ⊗ 1) = a for all a ∈ A, so ϕ is surjective. Since ϕ is a morphism of left (A ⊗2 * S 2 )-modules, to complete the proof, it is enough to show ϕ is an isomorphism of R-modules. Consider the R-module morphism 1) -(bc) (1) for all b, c ∈ A, that is, G(A/R) is generated as an R-module by the image of elements of the form a (1) for a ∈ A.
Remark 3.3. Proposition 3.2 generalizes the fact that a separable degree 2 field extension L/K is already Galois and hence its Galois closure is L. Note that in the case of quadratic R-algebras, since G(A/R) A, the Galois closure inherits a ring structure.
We give some examples of G(A/R) where A has degree 3 but is the product of smaller degree algebras. These Galois closures may be easily computed using the product formula (Theorem 2.26), but in this section, we show how to work with them explicitly.
The simplest case of a decomposable degree 3 R-algebra is A = R × R × R. Each element a = (r 1 , r 2 , r 3 ) ∈ A satisfies the polynomial
where the trace Tr(a) is s A,1 (a) = t = r 1 + r 2 + r 3 , the spur Spr(a) is s A,2 (a) = s = r 1 r 2 +r 1 r 3 +r 2 r 3 , the norm N(a) is s A,3 (a) = n = r 1 r 2 r 3 , and 1 A denotes the multiplicative identity element (1, 1, 1) in A.
We claim that G(A/R) is isomorphic to R ⊕6 as left (A ⊗3 * S 3 )-modules, where the left (A ⊗3 * S 3 )-module structure on R ⊕6 is given as follows. We index each of the six copies of R in R ⊕6 by the six permutations of {1, 2, 3}. The three actions of (r 1 , r 2 , r 3 ) ∈ A on (c ijk ) {i,j,k}={1,2,3} ∈ R ⊕6 are as follows:
(r 1 , r 2 , r 3 ) • 1 (c ijk ) {i,j,k}={1,2,3} := (r i c ijk ) {i,j,k}={1,2,3}
(r 1 , r 2 , r 3 ) • 2 (c ijk ) {i,j,k}={1,2,3} := (r j c ijk ) {i,j,k}={1,2,3}
(r 1 , r 2 , r 3 ) • 3 (c ijk ) {i,j,k}={1,2,3} := (r k c ijk ) {i,j,k}={1,2,3} . The action of σ ∈ S 3 on R ⊕6 is the standard action on the indices, that is, σ ((c ijk ) ijk ) = (c σ (i),σ (j),σ (k) ) ijk . Then we define the morphism ϕ : A ⊗3 → R ⊕6 of left A ⊗3 -modules by linearly extending
It is easy to check that the image of I A/R is 0 in R ⊕6 , so we obtain an induced map
Proposition 3.5. The map ϕ is an isomorphism of left (A ⊗3 * S 3 )-modules:
Proof. This is a special case of Proposition 3.6 below, with B = R × R.
We next consider the more general situation where B is a quadratic R-algebra We claim that the Galois closure G(A/R) is isomorphic to B ⊕3 where we endow B ⊕3 with a left (A ⊗3 * S 3 )-module structure as follows. The three actions of (r, c) ∈ A on
The S 3 -action on B ⊕3 is given by Then we may define a morphism ϕ :
An easy check shows that ϕ(I A/R ) = 0, so we obtain an induced map ϕ :
We then have the following.
Proposition 3.6. The map ϕ is an isomorphism of left (A ⊗3 * S 3 )-modules:
Proof. We see that ϕ is surjective since ϕ is: for every b ∈ B, we have
Since we know ϕ is a surjective map of left (A ⊗3 * S 3 )-modules, to prove it is an isomorphism, it is enough to show it is an isomorphism of R-modules. To do so, we need only find an R-module map ψ : B ⊕3 → G(A/R) that is a surjective section of ϕ. Then we can define ψ by ψ(b, 0, 0) = (0, 1) (2) (0, b) (3) , ψ(0, b, 0) = (0, b) (3) -(0, 1) (2) (0, b) (3) , and (2) -(0, 1) (2) (0, b) (3) . It then suffices to show that G(A/R) is generated as an R-module by elements of the form (0, b) (2) , (0, b) (3) , and (0, 1) (2) (0, b) (3) .
For any element α ∈ A, the trace relation α (1) + α (2) + α (3) -Tr(α) ∈ I A/R trivially shows that any element α (1) = α ⊗ 1 ⊗ 1 ∈ A ⊗3 may be written in G(A/R) as an Rlinear combination of 1 A , α (2) , and α (3) . In particular, any element of G(A/R) may thus be written as a linear combination of elements of the form β (2) γ (3) for β, γ ∈ A. Also,
, so the Galois closure G(A/R) is generated as an R-module by 1 A and elements of the form (0, b) (2) , (0, b) (3) , and (0, b) (2) (0, b ) (3) for b, b ∈ B.
We claim that for b, b ∈ B, we can write (0, b) (2) (0, b ) (3) in the form (0, 1) (2) (0, b ) (3) Next, the trace and spur relations tell us
where the last equality holds because Tr A (0, 1) = Tr B (1 B ) = 2. Applying this equation once to left-hand side and twice to the right-hand side of
and making use of (3.7) implies
that simplifies to
(0, b) (3) .
Acting on the left by (0, b ) (3) shows that (0, b) (2) (0, b ) (3) = (0, 1) (2) (0, b b) (3) , that is, we have shown that (0, b) (2) (0, b ) (3) is of the form (0, 1) (2) (0, b ) (3) .
So far, we have shown that G(A/R) is generated as an R-module by 1 A and elements of the form (0, b) (2) , (0, b) (3) , and (0, 1) (2) (0, b ) (3) . It remains to remove 1 A from our generating set. We have shown above that (0, b) (2)
Therefore, 1 A is also a linear combination of elements of the form (0, b) (2) , (0, b) (3) , and (0, 1) (2) (0, b) (3) . This concludes the proof.
Remark 3.9. As a consequence of Proposition 3.6, we see that G(A/R) inherits a ring structure when A = R×B with B a degree 2 R-algebra. This is not true more generally for indecomposable degree 3 R-algebras, for example, for the case A = Mat 3 (R) considered in Section 3.4. Let V be a free rank n module over R and let A = End(V) be the ring of R-module endomorphisms. Then as in Example 2.8, we may view A as a degree n R-algebra where for each α ∈ A, the polynomial P A,α (T) is the characteristic polynomial of α viewed as an endomorphism of V; the trace and the norm of an endomorphism coincide with their usual definitions.
Via the canonical isomorphism A V ⊗ V * , we have an isomorphism A ⊗n V ⊗n ⊗ (V * ) ⊗n . The natural left (A ⊗n * S n )-module structure on A ⊗n then induces such a structure on V ⊗n ⊗ (V * ) ⊗n . Explicitly, σ ∈ S n acts on the pure tensors via
where v j ∈ V and f j ∈ V * . The i-th action of α ∈ A is given by acting on the i-th factor of V:
We next define a left (A ⊗n * S n )-module structure on V ⊗n ⊗ n (V * ) as follows. For
in other words, for a form ω ∈ n (V * ), we have
For α ∈ A, the i-th A-action is given by
We then obtain a map of left (A ⊗n * S n )-left modules given by.
We show that ϕ induces an isomorphism between G(A/R) and V ⊗n ⊗ n (V * ). First, we show that we may find a fairly simple basis for A.
Lemma 3.12. Let V be a free rank n R-module. Then the endomorphism ring End(V) has a basis β 1 , . . . , β n 2 over R where each β i is a linear operator that is diagonalizable over R.
Proof. After choosing an R-basis for V, we have an isomorphism End(V) Mat n (R).
Let e ij denote the matrix whose entries are all 0 except for a 1 in the (i, j)-th position.
Then for i = j, the matrix e ij + e ii is diagonalizable: if P = e ii + e ije jj + k =i,j e kk , then P -1 = P and P -1 (e ij + e ii )P = e ii . So, we can choose our desired basis to be the n elements e 11 , . . . , e nn as well as the n 2n elements e ij + e ii and e ji + e jj with i < j.
Theorem 3.13. Let V be a free rank n R-module and A = End(V). The map ϕ of (3.10) induces an isomorphism
Proof. We begin by showing that ϕ(I A/R ) = 0. That is, we show that for all α ∈ A,
By Proposition 2.20, it suffices to prove this as α ranges over a basis of A over R.
Combining this with Lemma 3.12, we may assume that α is diagonalizable over R.
Choose a basis u 1 , . . . ,
where the last equality again follows from (3.11).
Having now shown that ϕ(I A/R ) = 0, we obtain an induced map ϕ : G(A/R) → V ⊗n ⊗ n (V * ). Since π is surjective, ϕ is as well. Since ϕ is a map of (A ⊗n * S n )-modules, to prove it is an isomorphism, it is enough to show it is an isomorphism of R-modules.
To do so, we construct a surjective R-module map that is a section of ϕ.
After choosing a basis of V, we may identify A with Mat n (R) and use the notation e ij to indicate a matrix that is 0 in all entries except 1 in the (i, j)-th position. It is clear that G(A/R) is generated by (the image of) the elements (e i 1 j 1 ) (1) • • • (e i n j n ) (n) for 1 ≤ i k , j k ≤ n. We claim that for every function τ : {1, 2, . . . , n} → {1, 2, . . . , n}, we have the following equality in G(A/R):
) (1) • • • (e i n n ) (n) for τ ∈ S n 0, for τ / ∈ S n .
(3.14)
Let N τ = j e jτ (j) . Then because e i k k N τ = e i k τ (k) and (N τ ) (1)
we have
Since det(N τ ) vanishes for τ / ∈ S n and is equal to sgn(τ ) for τ ∈ S n , we have (3.14). Thus, G(A/R) is generated by the elements (e i 1 1 ) (1) γ ∈ F * , one may define an associative F-algebra
The Albert-Brauer-Hasse-Noether theorem combined with the Grunwald-Wang theorem implies that if F is a number field, then every central simple algebra over F is a cyclic algebra, that is, of the form (3.17
) has order exactly n (i.e., γ n ∈ N K/F (K * ) and
Now fix a central simple algebra A over F of the form (3.17). Since K splits A, we have an injection of A into A ⊗ F K Mat n (K); explicitly A may be identified with the subring of Mat n (K) consisting of elements of the form
Since A K Mat n (K), the central simple algebra A corresponds to a Galois action on Mat n (K), which we now describe explicitly. Fix an n-dimensional K-vector space V and basis u 1 , . . . , u n of V to identify End(V) = Mat n (K). Recall the notation that e ij (or e i,j ) refers to the n × n matrix whose only nonzero entry is a 1 in the (i, j)-th position. For all α ∈ K, the action of Gal(K/F) = σ is given by
In other words, σ acts as usual on K, and it adds 1 to both of the i and j indices, multiplying by γ or γ -1 whenever the i or j index, respectively, overflows. Written in
The induced action on V ⊗n ⊗ n V * = G(A K /K) given by Proposition 3.16 is as follows: let u n+1 := u 1 and ω
where r is the number of j such that i j = n.
Example 3.18. Let F/Q be a field that does not contain a square root of -1 and let
Since G(A/F) is the subspace of elements that are fixed by σ , it is explicitly given by the set of
with α, β ∈ K. On the other hand, A is a quadratic algebra over F, so we know from Proposition 3.2 that G(A/F) A, where the left (A ⊗2 * S 2 )-module structure on A is described in Section 3.2. The map
yields an explicit isomorphism between our two different descriptions of G(A/F) as a left (A ⊗2 * S 2 )-module.
An easy application of the product formula (Theorem
As indicated in the introduction, one of our motivations for studying noncommutative Galois closures is related to constructing "Hermitian" representations. We describe in We would like to study representations of algebraic groups that generalize tensor products of standard representations.
A simple explicit example is the space W of m × m matrices that are Hermitian with respect to a quadratic algebra A over R. Using the notation for quadratic algebras from Section 3.2, these are m×m matrices C = (c ij ) such that C = C t , where C denotes the entrywise conjugate of C, that is, c ij = c ji for all 1 ≤ i, j ≤ m. If A is a rank ρ free module over R, then W is a free module over R of rank m + ρm(m -1)/2. Moreover, the group GL m (A) naturally acts on W: for γ ∈ GL m (A), we define the action of γ by γ • C = γ Cγ , where γ is the entrywise conjugate of γ . For example, if A is simply R itself considered as a degree 2 algebra with polynomial P r (T) = (Tr) 2 , then we have recovered the space of symmetric m × m matrices over R with the standard action of GL m (R); for A = R 2 as a quadratic algebra, this space is isomorphic to the space of m × m matrices over R with the standard action of GL m (R) × GL m (R). It is of course possible to define W in a basis-free manner, as we will see in more generality below.
We wish to generalize the above to a notion of a Hermitian n-dimensional m × • • • × m array with entries in a degree n algebra A over R. Intuitively, we would like the symmetric group S n to act on such an array in two different ways: by an S naction on A and by permuting the factors, and we would like to restrict to the arrays for which these two actions agree. In general, an algebra A does not come equipped with a natural S n -action, but its Galois closure G(A/R) does. So instead we allow for entries in G(A/R). We now make this definition precise.
We begin with a coordinate-free description of the representation and then give an explicit description in terms of coordinates. Let U be a free R-module of rank m.
There are two natural S n -actions on G(A/R) ⊗ R U ⊗n , one is the left action on G(A/R) and the other is a right action on U ⊗n given by permuting coordinates. For σ ∈ S n and ℵ ∈ G(A/R) ⊗ R U ⊗n , we denote the two actions by σ • 1 ℵ and σ • 2 ℵ, respectively. Definition 4.1. We define the Hermitian space H A,U to be the subspace of G(A/R) ⊗ R U ⊗n where the two S n -actions agree (up to an inverse), that is,
Remark 4.2. We have two commuting (left) S n -actions on G(A/R) ⊗ U ⊗n , hence an action of S n ×S n . The Hermitianization H A,U is the space of invariants for the diagonally embedded copy of S n in S n × S n , that is, H A,U = (G(A/R) ⊗ U ⊗n ) S n . Downloaded from https://academic.oup.com/imrn/article-abstract/doi/10.1093/imrn/rny231/5114641 by guest on 07 July 2020
Downloaded from https://academic.oup.com/imrn/article-abstract/doi/10.1093/imrn/rny231/5114641 by guest on 07 July 2020
In this section, we give many examples of Galois closures. Downloaded from https://academic.oup.com/imrn/article-abstract/doi/10.1093/imrn/rny231/5114641 by guest on 07 July 2020